I am Emily Josephs

• interested in: evolutionary genetics, plants, triathlons, cat

I am Emily Josephs

• interested in: evolutionary genetics, plants, triathlons, cat

1. Data into spreadsheet

2. Wrangle data into R

3. Visualize data

4. Build a model

   • how are data distributed?
   • relationship between predictor and response?

5. Use algorithm to parameterize model

6. Interpret results

1. Data into spreadsheet

2. Wrangle data into R

3. Visualize data

4. Build a model

   • how are data distributed?
   • relationship between predictor and response?

5. Use algorithm to parameterize model

6. Interpret results

Emily’s office hours:

noon-1pm Tuesday + Thursday in PLB 266

Sophie’s office hours:

11am-noon Monday + Wednesday in the EEB Hub and on Zoom

Simulations?

One of the most powerful tools that we’ll have in our statistics learning toolkit are simulations.

Simulations let you generate data that you know should look a certain way, so you can test your intuitions.

Simulations also let you do the same thing over and over and over.

If we randomly pick a student, how likely are we to select student #19?

If we randomly pick 10 students, how likely are we to select student #19?

If we randomly pick 2 students, how likely are we to select students #19 and #20?

If we randomly pick a student, how likely are we to select student #19?

mySamples <- replicate(10000, sample(thisClass, size=1))
sum(mySamples==19)/10000

## [1] 0.0304

The sample space for this example is that ( A), The ball can fall through the orange bin, ( B) the ball can fall through the green bin, and ( C) the ball can fall through the blue bin

Work with your groups to calculate the probability of A or B happening.

Note that this is called a complex event, since it is the combination of multiple, mutually-exclusive outcomes.

What if we want to simulate many samples to get a sampling distribution of how likely we are to get A or B?

nReps <- 100

sampleFunction <- function(myN){
                  mySample = sample(x=c("A","B","C"), 
                  size = myN,
                  replace=TRUE,
                  prob = c(pA, pB, pC))
                  
                  propA = sum(mySample=="A")/myN
                  propB = sum(mySample=="B")/myN
                  propC = sum(mySample=="C")/myN
                  
                  return(propA + propB)
                  }

mySamples <- replicate(nReps, sampleFunction(500))

hist(mySamples, main="", xlab = "proportion A or B")

How can we edit this sample function to return the probability of falling through both A and B if \(P(A) = 0.5\) and \(P(B) = 0.5\)?

sampleFunction <- function(myN){
                  mySample = sample(x=c("A","B","C"), 
                  size = myN,
                  replace=TRUE,
                  prob = c(pA, pB, pC))
                  
                  propA = sum(mySample=="A")/myN
                  propB = sum(mySample=="B")/myN
                  propC = sum(mySample=="C")/myN
                  
                  return(propA + propB)
                  }

pA = pB = 0.5

sampleFunction2 <- function(myN){
                  mySampleA = sample(x=c(1,0), 
                  size = myN, replace=TRUE,
                  prob = c(pA, 1-pA))
                  
                  mySampleB = sample(x=c(1,0), 
                  size = myN, replace=TRUE,
                  prob = c(pB, 1-pB))
                  
                  myCombined = mySampleA + mySampleB
                  
                  probBoth = sum(myCombined==2)/myN
                  
                  return(probBoth)
                  }

mySamples <- replicate(100,sampleFunction2(100))

hist(mySamples, main="", xlab = "proportion A and B")

Work with your teams to modify our sampling function to simulate the following scenario: \(P(A) = 1/3\), \(P(B|A) = 1/2\), \(p(B|not A) = 3/4\)

How often do we find A and B?

Note that this is called a shared event, made up of multiple non-exclusive outcomes.

pA = 1/2
pB_given_A    = 4/5
pB_given_notA = 1/5

#write a function to do one trial (one ball)

#write a function to get a sample by running that trial 500 times

#write a function to generate 100 samples

#write a function to do one trial (one ball)
oneTrial <- function(){
  mySampleA = sample(x=c(1,0), 
                  size = 1, replace=TRUE,
                  prob = c(pA, 1-pA))
                  
                  if (mySampleA == 1){pB = pB_given_A}
                  else{pB = pB_given_notA}
                  
                  mySampleB = sample(x=c(1,0), 
                  size = 1, replace=TRUE,
                  prob = c(pB, 1-pB))
                  
                  myCombined = mySampleA + mySampleB
                  isBoth = sum(myCombined==2)
                  return(isBoth)
}

#write a function to get a sample by running that trial 500 times

sampleFunction3 <- function(myN){sum(replicate(myN, oneTrial()))/myN}

#write a function to generate 100 samples
mySamples <- replicate(100,sampleFunction3(100))

hist(mySamples, main="", xlab = "Prob of A and B")